Group #26136847 — subgroups

Subgroups — #26136847

Order 12032.

184 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 105 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245682	C₂	1	2	yes	maximal, complemented
2	#20241586	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241530	C₂	1	2	yes	maximal, complemented
4	#20241522	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#2170	C₂	1	1	yes	maximal, characteristic
6	#15688957	—	2	2	no	—
7	#15688957	C₂²	1	2	yes	—
8	C₂×C₄₇⋊C₃₂	—	2	2	no	—
9	C₂×C₄₇⋊C₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₄₇⋊C₃₂	C₂²	1	1	yes	characteristic
11	C₄₇×#504	C₂²	1	2	yes	—
12	C₄₇×#443	C₄	1	1	yes	characteristic
13	C₄₇×#364	C₄	1	1	yes	characteristic
14	C₄₇×#350	C₄	1	2	yes	—
15	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇⋊C₃₂	—	2	2	no	—
17	C₄₇⋊C₃₂	—	2	1	no	—
18	C₄₇⋊C₃₂	D₄	1	2	yes	—
19	#135×C₄₇	C₈	1	1	yes	characteristic
20	#131×C₄₇	C₂×C₄	1	2	yes	—
21	M₃₂×C₄₇	—	2	2	no	—
22	M₃₂×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₄₇	—	2	2	no	—
24	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
26	#108×C₄₇	C₈	1	1	yes	characteristic
27	#104×C₄₇	C₈	1	2	yes	—
28	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	#41×C₄₇	C₂×C₈	1	2	yes	—
30	Q₁₆×C₄₇	C₁₆	1	1	yes	characteristic
31	SD₁₆×C₄₇	C₁₆	1	2	yes	—
32	D₈×C₄₇	C₁₆	1	1	yes	characteristic
33	M₁₆×C₄₇	—	2	2	no	—
34	M₁₆×C₄₇	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
39	C₁₆×C₄₇	—	2	2	no	—
40	C₁₆×C₄₇	—	2	1	no	—
41	C₁₆×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₄₇	#41	1	1	yes	characteristic
43	Q₈×C₄₇	C₂×C₁₆	1	2	yes	—
44	D₄×C₄₇	—	2	2	no	—
45	D₄×C₄₇	C₂×C₁₆	1	2	yes	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
49	C₈×C₄₇	—	2	2	no	—
50	C₈×C₄₇	C₂×C₁₆	1	1	yes	characteristic
51	C₈×C₄₇	C₂×C₁₆	1	1	yes	characteristic
52	C₈×C₄₇	C₄×D₄	1	1	yes	characteristic
53	C₈×C₄₇	#131	1	1	yes	characteristic
54	#13648	—	47	1	no	maximal, Sylow
55	C₂²×C₄₇	—	2	2	no	—
56	C₂²×C₄₇	#504	1	1	yes	characteristic
57	C₄×C₄₇	—	2	2	no	—
58	C₄×C₄₇	—	4	1	no	—
59	C₄×C₄₇	C₈×D₄	1	1	yes	characteristic
60	C₄×C₄₇	C₂²×C₁₆	1	1	yes	characteristic
61	#2258	—	47	2	no	—
62	#2170	D₄₇	1	1	yes	characteristic
63	#1422	—	47	1	no	—
64	#1401	—	47	2	no	—
65	#1397	—	47	1	no	—
66	C₂×C₄₇	—	2	1	no	—
67	C₂×C₄₇	—	4	2	no	—
68	C₂×C₄₇	D₄×C₁₆	1	1	yes	characteristic
69	#504	C₂×D₄₇	1	2	yes	—
70	#443	Dic₄₇	1	1	yes	characteristic
71	M₆₄	—	47	2	no	—
72	M₆₄	—	94	2	no	—
73	C₂×C₃₂	—	47	1	no	—
74	C₂×C₃₂	—	47	1	no	—
75	C₂×C₃₂	—	94	2	no	—
76	#364	Dic₄₇	1	1	yes	characteristic
77	#350	Dic₄₇	1	2	yes	—
78	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
79	C₄₇	#13648	1	1	yes	characteristic, complemented, Sylow
80	#135	C₄₇⋊C₈	1	1	yes	characteristic
81	#131	C₂×Dic₄₇	1	2	yes	—
82	M₃₂	—	2	2	no	—
83	M₃₂	C₂×Dic₄₇	1	2	yes	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
86	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
87	#108	C₄₇⋊C₈	1	1	yes	characteristic
88	#104	C₄₇⋊C₈	1	2	yes	—
89	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
90	C₃₂	—	47	2	no	—
91	C₃₂	—	94	2	no	—
92	C₃₂	—	94	1	no	—
93	#41	C₂×C₄₇⋊C₈	1	2	yes	—
94	Q₁₆	C₄₇⋊C₁₆	1	1	yes	characteristic
95	SD₁₆	C₄₇⋊C₁₆	1	2	yes	—
96	D₈	C₄₇⋊C₁₆	1	1	yes	characteristic
97	M₁₆	—	2	2	no	—
98	M₁₆	C₂×C₄₇⋊C₈	1	2	yes	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
101	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
102	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	—	2	1	no	—
105	C₁₆	D₄×D₄₇	1	1	yes	characteristic
106	C₁₆	#10625352	1	1	yes	characteristic
107	Q₈	C₂×C₄₇⋊C₁₆	1	2	yes	—
108	D₄	—	2	2	no	—
109	D₄	C₂×C₄₇⋊C₁₆	1	2	yes	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
115	C₈	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
116	C₈	D₄×Dic₄₇	1	1	yes	characteristic
117	C₈	#13094031	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	#15689540	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	4	1	no	—
122	C₄	D₄×C₄₇⋊C₈	1	1	yes	characteristic
123	C₄	C₂²×C₄₇⋊C₁₆	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	4	2	no	—
126	C₂	D₄×C₄₇⋊C₁₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245682

C₂

yes

maximal, complemented

#20241586

C₂

yes

maximal, characteristic, complemented

#20241530

C₂

yes

maximal, complemented

#20241522

C₂

yes

maximal, characteristic, complemented

C₄₇×#2170

C₂

yes

maximal, characteristic

#15688957

—

#15688957

C₂²

yes

—

C₂×C₄₇⋊C₃₂

—

C₂×C₄₇⋊C₃₂

C₂²

yes

characteristic

C₂×C₄₇⋊C₃₂

C₂²

yes

characteristic

C₄₇×#504

C₂²

yes

—

C₄₇×#443

C₄

yes

characteristic

C₄₇×#364

C₄

yes

characteristic

C₄₇×#350

C₄

yes

—

C₄×C₁₆×C₄₇

C₂²

yes

characteristic

C₄₇⋊C₃₂

—

C₄₇⋊C₃₂

—

C₄₇⋊C₃₂

D₄

yes

—

#135×C₄₇

C₈

yes

characteristic

#131×C₄₇

C₂×C₄

yes

—

M₃₂×C₄₇

—

M₃₂×C₄₇

C₂×C₄

yes

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

#108×C₄₇

C₈

yes

characteristic

#104×C₄₇

C₈

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

#41×C₄₇

C₂×C₈

yes

—

Q₁₆×C₄₇

C₁₆

yes

characteristic

SD₁₆×C₄₇

C₁₆

yes

—

D₈×C₄₇

C₁₆

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂×D₄

yes

characteristic

C₁₆×C₄₇

#41

yes

characteristic

Q₈×C₄₇

C₂×C₁₆

yes

—

D₄×C₄₇

—

D₄×C₄₇

C₂×C₁₆

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

C₂×C₁₆

yes

characteristic

C₈×C₄₇

C₂×C₁₆

yes

characteristic

C₈×C₄₇

C₄×D₄

yes

C₈×C₄₇

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

#504

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₈×D₄

yes

characteristic

C₄×C₄₇

C₂²×C₁₆

yes

characteristic

#2258

—

#2170

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

D₄×C₁₆

yes

characteristic

#504

C₂×D₄₇

yes

—

#443

Dic₄₇

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#364

Dic₄₇

yes

characteristic

#350

Dic₄₇

yes

—

C₄×C₁₆

C₂×D₄₇

yes

characteristic

C₄₇

#13648

yes

characteristic, complemented, Sylow

#135

C₄₇⋊C₈

yes

characteristic

#131

C₂×Dic₄₇

yes

—

M₃₂

—

M₃₂

C₂×Dic₄₇

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₄₇

yes

characteristic

C₂×C₁₆

C₂²×D₄₇

yes

characteristic

#108

C₄₇⋊C₈

yes

characteristic

#104

C₄₇⋊C₈

yes

—

C₄×C₈

C₂×Dic₄₇

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

#41

C₂×C₄₇⋊C₈

yes

—

Q₁₆

C₄₇⋊C₁₆

yes

characteristic

SD₁₆

C₄₇⋊C₁₆

yes

—

D₈

C₄₇⋊C₁₆

yes

characteristic

M₁₆

—

M₁₆

C₂×C₄₇⋊C₈

yes

—

C₂×C₈

—

100

C₂×C₈

C₂×C₄₇⋊C₈

yes

characteristic

101

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

102

C₄²

C₂×C₄₇⋊C₈

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

D₄×D₄₇

yes

characteristic

106

C₁₆

#10625352

yes

characteristic

107

Q₈

C₂×C₄₇⋊C₁₆

yes

—

108

D₄

—

109

D₄

C₂×C₄₇⋊C₁₆

yes

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×C₄₇⋊C₈

yes

characteristic

113

C₈

—

114

C₈

C₂×C₄₇⋊C₁₆

yes

characteristic

115

C₈

C₂×C₄₇⋊C₁₆

yes

characteristic

116

C₈

D₄×Dic₄₇

yes

characteristic

117

C₈

#13094031

yes

characteristic

118

C₂²

—

119

C₂²

#15689540

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

D₄×C₄₇⋊C₈

yes

characteristic

123

C₄

C₂²×C₄₇⋊C₁₆

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

D₄×C₄₇⋊C₁₆

yes

characteristic

	subgroup	G/H
center	C₁₆	D₄×D₄₇
commutator	C₄×C₄₇	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄₇×#2170	C₂
socle	C₂×C₄₇	D₄×C₁₆
prime radical	#135×C₄₇	C₈
cosocle	C₂×C₁₆×C₄₇	C₂³
hypercenter	#2170	D₄₇
hypocenter	C₄₇	#13648
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

subgroup

G/H

center

C₁₆

D₄×D₄₇

commutator

C₄×C₄₇

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₄₇

Fitting

C₄₇×#2170

C₂

socle

C₂×C₄₇

D₄×C₁₆

prime radical

#135×C₄₇

C₈

cosocle

C₂×C₁₆×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

N	H ≅ G/N	type
1. #20245682	C₂	semidirect
2. #20241586	C₂	semidirect
3. #20241530	C₂	semidirect
4. #20241522	C₂	semidirect
79. C₄₇	#13648	semidirect

H ≅ G/N

type

1. #20245682

C₂

semidirect

2. #20241586

C₂

semidirect

3. #20241530

C₂

semidirect

4. #20241522

C₂

semidirect

79. C₄₇

#13648

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20245682	2
#20241586	2
#20241530	2
#20241522	2
C₄₇×#2170	2
#13648	47

subgroup

index

C₄₇×#2170

Subgroups — #26136847

Conjugacy classes of subgroups

Complements

Maximal subgroups